[Paper Review] Projective Tensor Products of $C^*$-algebras

Ajay Kumar, Vandana Rajpal|arXiv (Cornell University)|May 3, 2013

Advanced Operator Algebra Research17 references4 citations

TL;DR

This paper investigates the projective tensor products of C*-algebras, focusing on the bi-continuity and isomorphism of the canonical embedding from $A^{**} \otimes_{\gamma} B^{**}$ into $(A \otimes_{\gamma} B)^{**}$, improving the operator norm bound to $\frac{1}{2}\|u\|_{\gamma} \leq \|\theta(u)\| \leq \|u\|_{\gamma}$. It further establishes that if one C*-algebra has finitely many closed ideals, then every closed ideal in $A \widehat{\otimes} B$ is a finite sum of product ideals, and proves the existence of outer automorphisms in $V \otimes_h W$ under certain conditions.

ABSTRACT

For $C^*$-algebras $A$ and $B$, we study the bi-continuity of the canonical embedding of $A^{**}\ot_γ B^{**}$ ($A^{**}\hat{\ot} B^{**}$) into $(A \ot_γ B)^{**}$ (resp. $(A \hat{\ot} B)^{**}$), and its isomorphism. Ideal structure of $A\hat{\ot} B$ has been obtained in case $A$ or $B$ has only finitely many closed ideals.

Motivation & Objective

• To analyze the bi-continuity and isomorphism of the canonical embedding $A^{**} \otimes_{\gamma} B^{**} \to (A \otimes_{\gamma} B)^{**}$ for C*-algebras $A$ and $B$.
• To characterize the ideal structure of the operator space projective tensor product $A \widehat{\otimes} B$ when one of the algebras has finitely many closed ideals.
• To investigate the existence of outer automorphisms in $A \otimes_h B$ and $A \widehat{\otimes} B$ for C*-algebras and exact operator algebras.
• To extend spectral and homomorphism properties of slice maps to the Haagerup tensor product setting.

Proposed method

• Uses the duality between the Banach space projective tensor norm and the injective tensor norm to re-derive the bi-continuity of the canonical embedding with an improved constant.
• Applies the non-commutative Grothendieck theorem and recent results on jointly completely bounded bilinear forms to analyze the operator space projective tensor product.
• Employs slice maps and pure states to relate spectral properties in $A \otimes_h B$ to those in the factor algebras.
• Utilizes the Haagerup norm and its properties to study automorphisms in $V \otimes_h W$, particularly via the module property of slice maps.
• Applies Corollary 4.5 and Proposition 4.4 to show injectivity and surjectivity of tensor maps in the Haagerup setting.
• Uses the relative commutant $V \otimes_h Z(W)$ and the structure of inner automorphisms to prove the existence of outer automorphisms when $\Phi$ is outer.

Experimental results

Research questions

• RQ1 Under what conditions is the canonical embedding $A^{**} \otimes_{\gamma} B^{**} \to (A \otimes_{\gamma} B)^{**}$ an isomorphism?
• RQ2 What is the optimal constant in the norm inequality $\frac{1}{2}\|u\|_{\gamma} \leq \|\theta(u)\| \leq \|u\|_{\gamma}$ for the canonical map $\theta$?
• RQ3 When does every closed ideal in $A \widehat{\otimes} B$ decompose as a finite sum of product ideals?
• RQ4 Can the existence of outer automorphisms in $V \otimes_h W$ be guaranteed when $V$ has a completely contractive outer automorphism and $W$ is $*$-reduced?
• RQ5 How do spectral properties of elements in $A \otimes_h B$ relate to those in the factor algebras via slice maps?

Key findings

• The canonical map $\theta: A \otimes_{\gamma} B \to (A^* \otimes_{\lambda} B^*)^*$ satisfies $\frac{1}{2}\|u\|_{\gamma} \leq \|\theta(u)\| \leq \|u\|_{\gamma}$, improving the previous bound and proving bi-continuity.
• If $A$ or $B$ has only finitely many closed ideals, then every closed ideal in $A \widehat{\otimes} B$ is a finite sum of product ideals, a result that fails for the minimal tensor product.
• For unital operator algebras $V$ and $W$ with $W$ $*$-reduced and $V$ having a completely contractive outer automorphism $\Phi$, the map $\mu(x) = \Phi(a_i) \otimes b_i$ on $V \otimes_h W$ is a completely contractive outer automorphism.
• The slice map $R_\phi: A \otimes_h B \to B$ is an algebra homomorphism when $A$ is a unital commutative $*$-reduced Banach $*$-algebra and $\phi$ is a pure state.
• If $\sigma(ux) = \sigma(vx)$ for all $x \in A \otimes_h B$, then $u = v$ in $A \otimes_h B$ for unital commutative $*$-reduced $A$ and unital $B$, due to spectral equality via slice maps.
• The map $\phi \otimes_h \psi$ is an isomorphism of $V \otimes_h W$ onto itself when $\phi$ and $\psi$ are automorphisms of $V$ and $W$, respectively, and is injective and surjective in the Haagerup tensor product.